CHEMICAL ENGINEERING

6 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Solution

The flowrate,Q, will be a function of the fluid density, , and viscosity, ,thefilm
thickness,d, and the acceleration due to gravity,g,

or: QDf
,g,,d,or:QDK
agb cddwhereKis a constant.

The dimensions of each variable are:QDL^2 /T, DM/L^3 ,gDL/T^2 , DM/LT
anddDL.
Equating dimensions:

M: 0 DaCc

L: 2 D 3 aCbcCd

T: 1 D 2 bc

from which,cD 1  2 b,aDcD 2 b 1 ,anddD 2 C 3 abCc
D 2 C 6 b 3 bC 1  2 bD 3 b

∴ QDK
^2 b^1 gb 1 ^2 bd^3 b

or:

Q

DK

^2 gd^3 /^2 bandQ/ 1 ^2 b.

For streamline flow,Q/ ^1

and:  1 D 1  2 bandbD 1

∴ Q
/DK
^2 gd^3 /^2 ,QDK
gd^3 /

and: Qis directly proportional to the density,

PROBLEM 1.

Obtain, by dimensional analysis, a functional relationship for the heat transfer coefficient
for forced convection at the inner wall of an annulus through which a cooling liquid is
flowing.

Solution

Taking the heat transfer coefficient,h, as a function of the fluid velocity, density, viscosity,
specific heat and thermal conductivity,u, , ,Cpandk, respectively, and of the inside
and outside diameters of the annulus,diandd 0 respectively, then:

hDfu,di,d 0 ,

,,Cp,k

The dimensions of each variable are:hDH/L^2 Tq,uDL/T,diDL,d 0 DL, DM/L^3 ,
DM/LT,CpDH/Mq,kDH/LTq. There are 8 variables and 5 fundamental dimen-
sions and hence there will be 8  5 D3 groups.Handqalways appear however as